DNA Implementation of Theorem Proving with

Resolution Refutation in Propositional Logic

In-Hee Lee1, Ji-Yoon Park2, Hae-Man Jang2, Young-Gyu Chai2, andByoung-Tak Zhang1

1 Biointelligence LaboratorySchool of Computer Science and Engineering

Seoul National University, Seoul 151-742, Korea2 Department of Biochemistry and Molecular BiologyHanyang University, Ansan, Kyongki-do 425-791, Korea{ihlee,jypark,hmjang,ygchai,btzhang}@bi.snu.ac.kr

Abstract. Theorem proving is a classical AI problem having a broadrange of applications. Since its complexity grows exponentially with thesize of the problem, many researchers have proposed methods to par-allelize the theorem proving process. Here, we use the massive paral-lelism of molecular reactions to implement parallel theorem provers. Inparticular, we show that the resolution refutation proof procedure canbe naturally and efficiently implemented by DNA hybridization. NovelDNA encoding schemes, i.e. linear encoding and hairpin encoding, arepresented and their effectiveness is verified by biochemical experiments.

1 Introduction

DNA computing is famous for its power of massive parallelism. Since Adle-man’s first experiment [1], many researchers utilized parallel reactions of DNAmolecules to solve hard computational problems [3,7,18]. Recently, several re-search groups have proposed DNA computing methods for logical reasoning[6,10,16,17].

Theorem proving is a method for logical reasoning and has a variety of ap-plications, including diagnosis and decision making [8,11]. Resolution refutationis a general technique to prove a theorem given a set of axioms and rules. Buttheorem proving by resolution refutation has a difficulty in practice. If the goalbecomes complex or the number of axioms gets large, the time for theorem prov-ing grows exponentially. To overcome this drawback, parallel theorem provershave been proposed [5,9,15]. However, these parallel machines do not overcomethe difficulties inherent to silicon-based technology.

Wasiewicz et al. [17] describe an inference system using molecular comput-ing. Their inference system is different from ours in that theirs does not use aresolution refutation technique. A resolution method for Horn clause computa-tion was suggested in [6,16], but was not used for theorem proving. Mihalacheproposed an implementation of Prolog interpreter with DNA molecules whichis an important practical application of resolution refutation theorem proving

M. Hagiya and A. Ohuchi (Eds.): DNA8, LNCS 2568, pp. 156–167, 2003.c© Springer-Verlag Berlin Heidelberg 2003

DNA Implementation of Theorem Proving 157

[10]. But his suggestion is not physically feasible, because it requires too manyexperimental steps. Except for the inference system in [17], none of these wasimplemented in real bio-lab experiments.

In this paper, we describe resolution refutation theorem proving methods us-ing DNA, which are verified by experiments. We develop two different encodingmethods for logical formulas, i.e. linear and hairpin encodings. These make useof the DNA hybridization reaction in a natural way to perform resolution refu-tation. Our implementation requires only a constant number of lab steps. Thefeasibility of the methods is confirmed by lab experiments.

The rest of the paper is organized as follows. A brief introduction to theoremproving and resolution refutation is given in Section 2. Sections 3 and 4 describethe linear and hairpin implementation methods and their experimental results,respectively. Conclusions are drawn in Section 5.

2 Theorem Proving with Resolution Refutation

Theorem proving is a method for automated reasoning [8,11]. In theorem prov-ing, one must decide methods for representing information and inference rulesfor drawing conclusions [8]. Here, we confine ourselves to propositional logic.We use the resolution as inference rule. In this section, we will briefly describepropositional logic, resolution principle, and resolution refutation.

Propositional logic formula consists of Boolean variables and logical connec-tives. Boolean variable is a variable which can take only T (true) or F (false)as its value. Among basic logical connectives, there are ∧ (and, logical product),∨ (or, logical sum), ¬ (not, negation), and → (implication). A Boolean variableor its negation is called a literal. More exactly, a Boolean variable with ¬ con-nective is called a negative literal and one without it is called a positive literal.It is proven [2] that any n-ary connective can be defined using only ∧, ∨, and ¬.For example, A → B can be defined as ¬A ∨ B for any Boolean variables A, B.

To prove theorems using resolution refutation, every formula must be ex-pressed in clause form. A clause form in propositional logic is defined as follows:

(clauseform) := (clause) ∧ (clause) ∧ · · · ∧ (clause)(clause) := (literal) ∨ (literal) ∨ · · · ∨ (literal)

A clause with no literal is called an empty clause.Now the resolution principle can be defined. Let A and B be clauses, and v

be a literal such that v ∈ A and ¬v ∈ B. Then, from both A and B we can draw(A−{v})∨ (B−{¬v}). We say that we resolved A and B on v and the productof resolution is called a resolvent.

In general, a resolution refutation for proving an arbitrary formula ω from aset of formulas ∆ proceeds as follows [11]:

1. Put the formulas in ∆ into the clause form.2. Add the negation of ω, in clause form, to the set of clauses and call it C.

158 In-Hee Lee et al.

Fig. 1. Theorem proving using resolution refutation.

3. Resolve these clauses together, producing a resolvent that logically followsfrom them.

4. If an empty clause is produced, a contradiction is occurred. Thus, it is provedthat ω is consistent with ∆. Stop.

5. If no new resolvent can be produced, ω is proved not to be consistent with∆. Stop.

6. Else, add the resolvent to C and go to step 3.

When an empty clause is drawn, C is called the proof of the goal ω. Fig. 1shows an example of theorem proving process. The set of formulas ∆ = {P∧Q →R, S∧T → Q, S, T , P } is given. We want to prove R is consistent with ∆. Afterconverting the formulas into clause form, we get a set of clauses {¬P ∨ ¬Q∨R,¬S ∨ ¬T ∨ Q, S, T , P}. Then we add the negation of R, i.e. ¬R, to this set.Each box in Fig. 1 contains one clause. The clauses and their resolvents areconnected with arrows. Resolving on P from ¬P ∨ ¬Q ∨ R and P results in¬Q ∨ R. Similar steps can be continued until an empty clause is produced. Thesymbol nil denotes an empty clause.

The above theorem proving process becomes complex as the number of for-mulas grows. In the case of the propositional calculus, one must decide the literalto resolve on. Therefore, if there are n different literals, there are n! differenttheorem proving processes with n! = O(nn) by Stirling’s approximation. Thus,the complexity of proof grows exponentially. So it is impossible for large n totest one by one with digital computer which one is a logically correct proof. Tospeed up finding proofs, two approaches were developed. One approach is to useheuristics such as the breadth-first strategy, the set of support strategy, or theunit preference strategy [8]. The other approach is to parallelize the theoremproving process [5,9,15]. We took the second approach and chose to use the mas-sive parallelism of DNA molecular reactions for implementing parallel theoremprovers.

DNA Implementation of Theorem Proving 159

ACGTTAGA

TCTAACGT

TCGTCAGC

1. Represent Clauses with DNA1. Represent Clauses with DNA

2. Hybridization2. Hybridization

3. Ligation3. Ligation

4. Polymerase Chain Reaction (PCR)4. Polymerase Chain Reaction (PCR)

5. Gel Electrophoresis5. Gel Electrophoresis

(a) (b)

A

¬A

B

Fig. 2. (a) Encoding for a Boolean variable and its negation (The arrows arefrom 5’ to 3’). (b) The general procedure for our experiments.

3 Linear Implementation of Resolution Refutation

We solved the theorem proving problem shown in Fig. 1 by biochemical experi-ments. We developed two versions of implementation and performed experimentsseparately. Each implementation consists of two steps.

In the first step, we represent formulas in clause form with DNA molecules.Because we restrict ourselves to propositional logic, we just need to encode eachBoolean variable with different DNA sequences. Our implementations are differ-ent from each other in the way to make a clause from these variable sequences.In both implementations, we encoded each variable with different sequences ofthe same length. The negation of a variable is denoted as a Watson-Crick com-plementary sequence encoding the variable. Encoding for a Boolean variable andits negation is shown in Fig. 2. In the second step, we implemented resolutionrefutation steps with molecular reactions. This step varies, depending on therepresentation method used in the previous step. But the general procedure isidentical in our two implementations as follows (see Fig. 2-(b)).

1. Mix DNA molecules corresponding to clauses.2. Hybridize DNA molecules to perform resolution.3. Ligate hybridization products to make it easy to find a proof.4. Perform PCR to amplify ligation products. In this step, only the ligation

product which is a valid proof can be amplified.5. Perform gel electrophoresis to see whether we found a proof or not.

3.1 Representation of Clauses

A clause is designed with single-stranded DNA that consists of each variablesin the clause. We determined the order in which a literal in a clause appears sothat a valid proof make a linear double-stranded molecule after hybridizationstep. As an example, sequence for a clause ¬Q ∨ ¬P ∨ R is a concatenation ofsequences for ¬Q, ¬P , and R (see the topmost box in Fig. 3).

160 In-Hee Lee et al.

SCTCA

SCTCA

PACGT

PACGT

TGTTA

TGTTA

¬RTGAC

¬RTGAC

¬Q ¬P RGACT TGCA ACGT¬Q ¬P R

GACT TGCA ACGT¬S ¬T Q

GAGT CAAT CTGA¬S ¬T Q

GAGT CAAT CTGA

¬S ¬T QGAGT CAAT CTGA

¬S ¬T QGAGT CAAT CTGA ¬P R

GACT TGCA ACGT¬P R

GACT TGCA ACGTSCTCA

SCTCA

PACGT

PACGT

TGTTA

TGTTA

¬RTGAC

¬RTGAC

¬S ¬T QGAGT CAAT CTGA

¬S ¬T QGAGT CAAT CTGA

GACT TGCA ACGTGACT TGCA ACGTCTCACTCA

PACGT

PACGT

GTTAGTTA

¬RTGAC

¬RTGAC

Fig. 3. Simplified process of linear implementation (The arrows are from 5’ to3’).

3.2 Implementation of Resolution Refutation

Resolution of a variable between two clauses is represented with hybridizationof two regions corresponding to that literal in each clause. For example, whena variable v is resolved from clauses A and B, regions corresponding to v or¬v in each clause hybridize and other regions remain unchanged. Therefore, ifthe resolvent is an empty clause, no region will remain as single-stranded DNA.Therefore, to see whether an empty clause is produced, one needs to verify if thereexists a molecule with no single-stranded region. We used ligation, PCR, andgel electrophoresis to find this molecule. During ligation, every clause sequenceused to produce the empty clause will be ligated into a long double-strandedmolecule. In the next step, we amplify this ligation product by PCR with thegoal sequence as a primer. Finally, the result is examined by gel electrophoresis.All of these experimental steps are summarized in Fig. 3.

3.3 Experimental Results

The sequences used in the experiment were designed by NACST using evolu-tionary optimization [14] and synthesized by Bioneer Co. (Tae-Jeon, Korea).The oligomer sequences are given in Table 1. The experiment consists of thefollowing steps:

1. Purification of oligomers: Each oligomer was 5’-phosphorylated and pu-rified by PAGE. Briefly, 1 nM of each oligomer was mixed and incubatedfor 1 h at 37◦C with 10 U T4 polynucleotide kinase (Life Technologies) in70 mM Tris-HCl (pH 7.6) buffer containing 10 mM MgCl2, 100 mM KCl, 1mM 2-mercaptoethanol and 1 mM ATP (Sigma, St. Louis, MO, USA), in avolume of 100 µl. The T4 kinase was inactivated by heating to 95◦C for 10min.

DNA Implementation of Theorem Proving 161

587587540540504504458458434434

124/123124/123104104

89/8089/80

64/57/5164/57/5121/18/11/821/18/11/8

267267234234213213192192184184

MM 11 22

75 75 bpbp

Fig. 4. Electrophoretogram of the PCR product in the linear implementation.Lane 1: PCR products with S and ¬R as primers. Lane 2: PCR products with¬S and R as primers. Lane M is a size marker.

2. Hybridization of oligomers: 100 pM of each oligomer was mixed. Initialdenaturation was achieved at 95◦C for 10 min. During hybridization welowered temperature from 95◦C to 16◦C (1◦C / min) using iCycler thermalcycler (Bio-rad, USA).

3. Ligation of hybrid molecules: Ligation was achieved with T4 DNA lig-ase at 16◦C for overnight using iCycler thermal cycler. The reaction buffercontains 50 mM Tris-HCl (pH 7.8), 10 mM MgCl2, 5 mM DTT, 1 mM ATP,and 2.5 µg/ml BSA.

4. PCR amplification for ‘readout’: 50 µl PCR amplification contained 100pM of primer and template. The reaction buffer consists of 10mM Tris-HCl(pH 7.8) containing 50 mM KCl, 1.5 mM MgCl2, 0.1% Triton X-100, 0.2mM dNTP, and 1 U DNA Taq polymerase (Korea Bio-technology, Korea).PCR condition is given in Table 2.

5. Gel electrophoresis: Amplified PCR products were purified by electro-phoresis on a 15% polyacrylamide gel (30% acrylamide [29:1 acrylamide /bis (acrylamide)]). The running buffer consists of 100 mM Tris-HCl (pH8.3) 89 mM boric acid, and 2 mM EDTA. The sample buffer is XyleneCyanol FF tacking dye. Gels were run on a Bio-rad Model Power PAC 3000electrophoresis unit at 60 W (6V/cm) with constant power.

Table 1. Sequences for linear implementation (in order of from 5’ to 3’).

clause sequence

¬Q ∨ ¬P ∨ R CGTACGTACGCTGAA CTGCCTTGCGTTGAC TGCGTTCATTGTATGQ ∨ ¬T ∨ ¬S TTCAGCGTACGTACG TCAATTTGCGTCAAT TGGTCGCTACTGCTT

S AAGCAGTAGCGACCAT ATTGACGCAAATTGAP GTCAACGCAAGGCAG¬R CATACAATGAACGCA

162 In-Hee Lee et al.

Table 2. The PCR condition for linear implementation.

cycle denaturation (94◦C) annealing (58◦C) polymerization (72◦C)1 4 min 0.5 min 0.5 min

2 ∼ 26 0.5 min 0.5 min 0.5 min27 0.5 min 0.5 min 10 min

The electrophoretogram is given in Fig. 4. To make an empty clause, all of the 5variables must be resolved. Every time one variable is resolved, double-strandedregion with 15 bp is produced. Therefore, we can expect a 75 bp band will appearafter gel electrophoresis. As can be seen in Fig. 4, an empty clause is produced.Thus, we found a proof for the given problem.

3.4 Discussion

There are some points to be considered to expand this implementation. Firstof all, we should rearrange variables in each clauses to make an empty clauseform a linear double-stranded molecule. This requirement poses a limitation onthe type of clauses we can use. That is, some kind of clauses does not make alinear double-stranded molecule no matter how we rearrange the variables. Also,it is impossible to know in advance which sequence to use as a primer. Andthere are possibilities of false positive. Even if S and T does not exist in Fig.3, our implementation will make band with expected length. But we can reducethe possibilities of false positive by including exonuclease treatment step beforePCR step.

4 Hairpin Implementation of Resolution Refutation

As mentioned in previous section, there are several limitations in linear imple-mentation. To overcome these limitations, we introduce branched molecules andhairpin molecules to represent clauses. In this implementation, we can alwaysuse the negation of goal as a primer. Similar idea was introduced by Uejima andothers[16] for Horn clause computation. But their work was intended to performHorn clause computation not resolution refutation. Also in their work, molecularform of the empty clause is different from ours.

4.1 Representation of Clauses

Each clause with n literals is denoted by a branched molecule with n arms exceptfor the clause with one literal. Each n-arm has a sticky end corresponding toeach literal. Sticky ends for the positive and negative literals of one variable arecomplementary. For a clause with one literal which is not the goal, we representit with a hairpin molecule having a sticky end. We encode the goal clause with alinear single-stranded molecule as in the linear implementation described in the

DNA Implementation of Theorem Proving 163

Fig. 5. The simplified process of hairpin implementation (The arrows are from5’ to 3’).

Table 3. Sequences for hairpin implementation (in order of from 5’ to 3’ ).

clause sequence

P TATTAAGACTTCTTGTAGTCT

¬P ∨ Q TAATAAGGAATCATGTTCCT

¬Q CATGA

previous section. For example, if there were ¬Q∨¬P ∨R, S, and ¬R (the goal),the clause ¬Q ∨ ¬P ∨R is represented by a 3-armed branched molecule, S by ahairpin molecule, and ¬R by a single-stranded molecule (see Fig. 5).

4.2 Implementation of Resolution Refutation

As in the linear implementation, resolution between two clauses is implementedas hybridization between two molecules. When an empty clause is drawn, it willstart with a goal sequence and end with its negation, since clauses are eitherbranched molecules or hairpin molecules except for the goal. Therefore, at thePCR step, we used the negation of goal variable only as a primer. To read thePCR product, we used gel electrophoresis. If a band is formed, we can concludethat the goal is consistent with the given clauses. If not, we say that the goalis not consistent with them. Because each band corresponds to a proof, we canfind several different proofs at one time.

164 In-Hee Lee et al.

5mer

6mer

5mer

5mer 5mer

(a) (b) (c)P

P

¬P V Q

¬P

Q

¬Q

¬Q

Fig. 6. The form of molecules used in experiments for hairpin implementation.(The arrows are from 5’ to 3’.)

Table 4. The PCR condition for hairpin implementation

cycle denaturation (98◦C) annealing (58◦C) polymerization (72◦C)1 5 min 1 min 1 min

2 ∼ 26 1 min 1 min 3 min27 1 min 1 min 7 min

4.3 Experimental Results

To test our idea, we solved a very simple theorem proving problem: given Pand P → Q, is Q consistent with them? Putting it in clause form, we get theclauses {P,¬P ∨ Q,¬Q}. The form of molecules representing these clauses aregiven in Fig. 6. As in the previous experiment, all sequences were designed byNACST [14] and were synthesized by Bioneer Co. The sequences we used aregiven in Table 3. Experimental steps are the same as in the previous experiment.The main differences between two experiments were the PCR condition and thetype of gel used in gel electrophoresis. The PCR condition for this experimentis given in Table 4. We should have used ¬Q, the negation of the goal variable,as a primer in principle. But in this case, the sequence corresponding to it is tooshort to use it as a primer. Therefore, we used lower part of the molecule in Fig.6-(b) as a primer. We used 3% agarose gel at the gel electrophoresis step.

The electrophorestogram of hairpin implementation is given in Fig. 7. In Fig.7, we can see bands of 23bp and 46bp as expected in Fig. 6. From this result,we can say our method can find a proof if it exist. If there is no such proof, weshould fail to get band with that length. To verify this, we performed the sameexperiments without one or more molecules in Fig. 6 (see Fig. 8). Only lane 12in Fig. 8 contains all of them. Therefore a band must appear in lane 12 only.But we can see short bands in lane 2∼7. We think that short bands in lane 3∼7are formed by hybridization of two P s(see Fig. 6-(a)). And the longer bands inlane 2 seems to be formed by hybridization of P and ¬P ∨Q (see Fig. 6-(a) andFig. 6-(b)). But this longer band must be shorter than the band in lane 12, forit does not contain molecule ¬Q. To compare both lanes, we draw dashed lines

DNA Implementation of Theorem Proving 165

50 50 bpbp

25 25 bpbp23 23 bpbp

M 1 2 3 4 5 6 7 8 M

Fig. 7. The result of ligation mixture after PCR and electrophoresis. Lane 1-8:Ligation mixture under various conditions. Lane M: a 25 bp size marker. Thearrow indicates the 23 bp PCR product.

in Fig. 8. We find that the band in lane 2 is shorter than that of lane 12 as weexpected.

4.4 Discussion

In this experiment, because of self-complementary sequences such as hairpin andgroup of sequences with complementary subsequence such as branch, PCR stepis extremely difficult. If some of these complementary sequences hybridize duringPCR step due to their thermodynamical properties such as melting temperature,amplification will stop at that point and we will get false negative result. Alsoas you can see in Fig. 7, one proof can have two bands of different length. Wethink it is due to the hybridization of two hairpin molecules.

And there are some cases when our method can not tell whether a proof isfound or not. For example, if the goal variable is resolved more than once, ourmethod will tell there exists a proof regardless of the existence of empty clause.To solve this problem, we need a different detection method for empty clause ora different encoding scheme.

Also, for we did not restricted the form of clause, non-Horn clause can form acomplicated self-loop structure. And even if we restrict ourselves to Horn-clauses,if there are two clauses which have two resolvable variables, they can form aself-loop structure, too. For example, if there are ¬P ∨ Q and P ∨ ¬Q ∨ ¬R,after resolving P or Q, the resolvent can form a self-loop structure. What isworse, as the number of variable grows, we should use longer sequences to encodeeach variable and the possibility of self-hybridization will grow. To solve thisproblem, we need to make procedure removing this self-loop or to design newrepresentations for clauses.

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1 2 3 4 5 6 7 8 9 10 11 12 M

25 25 bpbp50 50 bpbp

Fig. 8. The verification of detection method. Lane 1: without Fig. 6-(a). Lane2: without Fig. 6-(c). Lane 3: without upper part of Fig. 6-(b). Lane 4: withoutlower part of Fig. 6-(b). Lane 5: Fig. 6-(a) and (c) only. Lane 6: Fig. 6-(a) andlower part of (b) only. Lane 7: Fig. 6-(a) and upper part of (b) only. Lane 8:Fig. 6-(b) only. Lane 9: lower part of Fig. 6-(b) and (c) only. Lane 10: upperpart of Fig. 6-(b) and (c) only. Lane 11: the same as lane 8. Lane 12: with all ofmolecules. Lane M: a 25bp size marker.

When using branched molecules, branch migration may occur. But as sug-gested in [12], inserting T’s to junction point can reduce the possibility of branchmigration.

5 Conclusions

Using molecular reactions of DNA, we proved theorems in the propositionalcalculus. We presented methods for encoding clauses with DNA molecules andsolved theorem proving problems with lab experiments. Our methods are dis-tinguished from other work in several points. First, it does not need additionaloperations except hybridization. Only simple operations such as ligation andPCR are needed to verify the results. Second, the number of experimental stepsdoes not vary with the problem size. Our implementation methods require onlyhybridization, ligation, PCR, and gel electrophoresis, and these operations areall O(1) operations. Finally, taking the limit of PCR operation (10,000bp) intoconsideration, our methods can solve theorem proving problems with up to 660literals (15mer per literal).

As we discussed above, there are several things to consider in both imple-mentations. And we are trying to improve our implementations.

Acknowledgements

This research was supported in part by the Ministry of Education under theBK21-IT program and the Ministry of Commerce through the Molecular Evo-

DNA Implementation of Theorem Proving 167

lutionary Computing (MEC) project. The RIACT at Seoul National Universityprovided research facilities for this study.

References

1. Adleman, L., Molecular computation of solutions to combinatorial problems, Sci-ence, 266:1021–1024, 1994.

2. Fitting, M., First-Order Logic and Automated Theorem Proving, Springer-VerlagNew York Inc., 1942.

3. Hagiya, M., Arita, M., Kiga, D., Sakamoto, K., and Yokoyama, S., Towards par-allel evaluation and learning of Boolean µ-formulas with molecules, PreliminaryProceedings of the Third DIMACS Workshop on DNA Based Computers, 105–114,1997.

4. Hagiya, M., From molecular computing to molecular programming, Lecture Notesin Computer Science, 2001.

5. Hasegawa, R., Parallel theorem-proving system: MGTP, Proceedings of Fifth Gen-eration Computer System, 1994.

6. Kobayashi, S., Horn clause computation with DNA molecules, Journal of Combi-natorial Optimization, 3:277-299, 1999.

7. Lipton, R.J., DNA solution of hard computational problem, Science, 268:542–545,1995.

8. Luger, G.F. and Stubblefield, W.A., Artificial Intelligence: Structures and Strate-gies for Complex Problem Solving, 2nd Ed., Benjamin/Cummings, 1993.

9. Lusk, E.L. and McCune, W.W., High-performance parallel the-orem proving for shared-memory multiprocessors, http://www-fp.mcs.anl.gov/∼lusk/papers/roo/paper.html, 1998.

10. Mihalache, V., Prolog approach to DNA computing, Proceedings of the IEEE In-ternational Conference on Evolutionary Computation, IEEE Press, 249–254, 1997.

11. Nilsson, N.J., Aritificial Intelligence: A New Systhesis, Morgan Kaufman Publish-ers Inc., 1998.

12. Sa-Ardyen, P., Jonoska, N., and Seeman, N.C., Self-assembling DNA Graphs, Pre-liminary Proceedings of the Eighth International Meeting on DNA Based Comput-ers, 20–28, 2002.

13. Sakamoto, K., Gouzu, H., Komiya, K., Kiga, D., Yokoyama, S., Yokomori, T., andHagiya, M., Molecular computation by DNA hairpin formation, Science, 288:1223-1226, 2000.

14. Shin, S.-Y., Kim, D., Lee, I.-H., and Zhang, B.-T., Evolutionary sequence gener-ation for reliable DNA computing, 2002 IEEE World Congress on EvolutionaryComputation, 2002 (accepted).

15. Suttner, C., SPTHEO - A parallel theorem prover, Journal of Automated Reason-ing, 18(2):253-258, 1997.

16. Uejima, H., Hagiya, M., and Kobayashi, S., Horn clause computation by self-assembly of DNA molecules, Preliminary Proceedings of the Seventh InternationalMeeting on DNA Based Computers, 63–71, 2001.

17. Wasiewicz, P., Janczak, T., Mulawka, J.J., and Plucienniczak, A., The inferencebased on molecular computing, International Journal of Cybernetics and Systems,31(3):283–315, 2000.

18. Winfree, E., Algorithmic self-assembly of DNA, Ph.D. Thesis, California Instituteof Technology, 1998.